The Open Operational Research Journal, 2008, 2, 1-7
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Comparing Fundamentals of Additive and Multiplicative Aggregation in Ratio Scale Multi-Criteria Decision Making E.U. Choo* and W.C. Wedley Faculty of Business Administration, Simon Fraser University, B.C., Canada Abstract: Additive and multiplicative aggregations of ratio scale preferences are frequently used in multi-criteria decision making models. In this paper, we compare the advantages and limitations of these two aggregation rules by exploring only their fundamental properties after ratio scaled local priorities and criteria weights have been successfully generated from the decision maker. The comparisons of these properties are therefore independent of ancillary procedures such as interactive elicitations from decision makers, pairwise comparisons and calculations of local priorities and criteria weights. We compare six fundamental properties of the two aggregation rules. The criteria weights used in the multiplicative aggregation have complicated meanings which are not well understood and often mixed up in the ambiguous notion of "criteria importance". As the scaling factors of the local preference values do not appear explicitly in the computations of the relative ratios of the overall preferences in the multiplicative aggregation model, the relative ratios remain unchanged when the scaling factors are changed or an alternative is added or deleted. Furthermore, the relative ratios in the multiplicative aggregation do not depend on similar local preference values which cancel each other out mathematically. It is quite evident that the additive aggregation model is superior and easier for decision makers to use and understand. We recommend the additive aggregation rule over the multiplicative aggregation rule.
Keywords: Preferences, ratio scale, unit of measure, criteria weights, additive aggregation, multiplicative aggregation. 1. INTRODUCTION The basic MCDM problem in most decision making is to evaluate competing alternatives under multiple conflicting criteria. Even though rank ordering (an ordinal scale) of the alternatives is the most common form of solution sought by most decision makers (DM), it is always desirable to know the relative standings of the alternatives measured on a scale containing more information. Valuation on a ratio scale is preferred because it provides the DM with a relative measure of alternatives on each criterion as well as the overall ratio preference across all criteria. A ratio scale measure of overall preference value is also useful for the allocation of resources among all the alternatives. With a ratio scale, it is meaningful to reach conclusions such as "alternative Aj is r times preferred to Ak relative to all criteria". Additive aggregation and multiplicative aggregation of the local ratio preferences of each alternative into an overall preference are frequently used in multi-criteria decision making models [1-3].
Fundamental basic elements of the MCDM framework are first depicted without any specific interpretations imposed on these elements. It is assumed that no relevant criterion is missed and each criterion is autonomous. In section 3, the measures of criteria weight, local and overall preferences are assumed to be in ratio scale. Some necessary conditions and the role of normalization are discussed. We then give a brief literature review, with particular attention to the different ancillary procedures and contradicting opinions in model interpretations. Additive and multiplicative aggregation rules are formally introduced in Section 5. In Section 6, we elaborate and compare the fundamental properties of these aggregation rules. Finally, we summarize and give some conclusions. 2. BASIC ELEMENTS OF MCDM MODEL
In this paper, we compare the advantages and limitations of additive aggregation and multiplicative aggregation rules by exploring their fundamental properties. We do this without investigating the different approaches to ancillary and peripheral procedures such as interactive elicitations from decision makers, pairwise comparisons, and calculations of local priorities and criteria weights. By assuming that local priorities and criteria weights have been correctly derived, we are able to focus on the aggregation procedures. From the comparisons of the fundamental properties presented, it is quite evident that the additive aggregation rule is superior with simpler interpretations which are more readily understood by decision makers.
The basic elements of a typical MCDM model include a set A={A1,A2,…,An} of n alternatives A1,A2,…,An and a set C={C1,C2,…,Cm} of m criteria C1,C2,…,Cm. The effect of the criteria C1,C2,…,Cm in C is represented by positive numbers w1,w2,…,wm respectively. The vector w=[w1,w2,…,wm] is called the criteria weight vector of the criteria C1,C2,…,Cm in C. The criteria weight vector w is derived from questioning the DM. The alternatives A1,A2,…,An can be evaluated under each individual criterion Cp, p=1,2,…,m. For each criterion Cp (p=1,2,…,m), the local preference of the alternatives A1,A2,…,An in A with respect to Cp is represented by positive numbers x1p,x2p,…,xnp, respectively. The vector xp=[x1p,x2p,…,xnp] is called the local preference vector of the alternatives A1,A2,…,An in A with respect to Cp. The local preference vectors x1,x2,…,xm are derived from questioning the DM.
*Address correspondence to this author at the Faculty of Business Administration, Simon Fraser University, B.C., Canada; E-mail:
[email protected]
It is important to note that at this rudimentary level, we only assume that the numerical values in the criteria weight vector w and the local preference vectors x1,x2,…,xm exist
1874-2432/08
2008 Bentham Science Publishers Ltd.
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and are used in MCDM model without stipulating any specific properties, interpretations and restrictions. The properties of the scale of these numerical values and the methods of evaluating them are regarded as additional model assumptions which could vary widely among different models [4, 5]. These numerical values can represent mea-sures in ordinal scale [6], interval scale [7] or ratio scale [8, 9]. We note that the outranking relationship [10] is not a complete order and thus may not be represented by such numerals directly. The ultimate goal of all MCDM problems is to aggregate the evaluation of the n alternatives with respect to the m criteria into overall preference measures v1,v2,…,vn of the alternatives A1,A2,…,An respectively. The vector v=[v1,v2,…,vn] is called the overall preference vector of the alternatives A1,A2,…,An in A with respect to all the criteria in C. The best alternative is the one with the largest overall preference value. The fundamental assumption in MCDM models is that the overall preference vector v=[v1,v2,…,vn] is a function of the criteria weight vector w=[w1,w2,…,wm] and the local preference vectors x1,x2,…,xm. Consider five houses A1,A2,A3,A4,A5 to be evaluated with respect to three criteria C1,C2,C3. The basic elements are shown in Table 1. For each criterion Cp, we first compute the vector of local preference xp=[x1p,x2p,x3p,x4p,x5p] (p=1,2,3). Then we determine the criteria weights w1,w2,w3. Finally the overall preference vector v=[v1,v2,v3,v4,v5] is derived by combining the local priority vectors x1,x2,x3 with the criteria weights w1,w2,w3 according to some aggregation rule. Table 1.
Basic Elements of MCDM: Alternatives A1,A2,A3,A4, A5 and Criteria C1,C2,C3
Alternatives
C1 Price
C2 Location
C3 Condition
Overall
House A1
x11
x12
x13
v1
House A2
x21
x22
x23
v2
House A3
x31
x32
x33
v3
House A4
x41
x42
x43
v4
House A5
x51
x52
x53
v5
Criteria Weight
w1
w2
w3
To avoid solving the wrong problem, we assume that all relevant criteria are included in C so that no new criterion will be added. To avoid redundancy, we assume that no criterion can be deleted and the criteria in C are disjoint with each other in the sense that no criterion is double counted by another criterion. Each criterion Cp (p=1,2,…,m) is assumed to be "autonomous" in the sense that the performance of each alternative Aj (j=1,2,…,n) with respect to Cp is independent of the other alternatives. In particular, none of the criteria in C is a measure of scarcity, abundance or some ranking of the alternatives in A. It follows that the evaluation of Aj (j=1,2,…,n) with respect to Cp remains the same regardless of whether any other alternative is deleted or added to A. In MCDM models, the numerical values in the local preference vectors x1,x2,…,xm and the criteria vector w are generated so that they can be aggregated to overall preferences. Some necessary measurement scale underlying the
numerical values must be established and maintained so that the aggregated results are meaningful measures. For example, the sum of ordinal numerical values such as rankings is not a valid measure of magnitudes. Nor is it meaningful to add or multiply numerical values in commensurate interval scale such as 25oC+34oC for temperature in Celsius. Thus, (a+bx1)+(a+bx2) = 2a+b(x1+x2) and (a+bx1)(a+bx2) = a2+ab(x1+x2)+b2x1x2 are not meaningful as interval scales because they do not have the form a+bx. However, the arithmetic average or weighted average of numerical values in commensurate interval scale is meaningful. For example, [(a+bx1)+(a+bx2)]/2 = a+b(x1+x2)/2 = a+b(x3) which is of the form a+bx. Similarly, a weighted average keeps the form a+bx if the sum of the weights equals 1 – e.g. 0.3(a+bx1)+0.7(a+bx2) = a+b(0.3x1+0.7x2) = a+b(x3). For ratio scales, the situation is different – both addition and multiplication are permissible. Thus, bx1+bx2 = b(x1+x2) = b(x3) belongs to the same ratio scale if they are commensurate. Note that ax1 and bx2 cannot be added because they are in different scales. For multiplication of values on the same ratio scale, (bx1)(bx2) = b2x1x2 = c(x3) which is a new ratio scale with a squared unit of measure. As well, multiplication of values from different ratio scales produces a new ratio scale: e. g. (ax1)(bx2) = abx1x2 = c(x3). Raising different ratio values to powers also produces a new ratio scale – e. g. (ax1)w(bx2)(1-w) = [awb(1-w)][x1wx2(1-w)] = c(x3). Unlike addition, multiplication of ratio scales produces a new ratio scale with a different unit of measure. 3. RATIO SCALE MEASURES AND SCALING FACTORS To be more focused in our investigation, we emphasize on ratio scales that allow both addition and multiplication for aggregation. We assume that the overall preferences v1,v2,…,vn exist in ratio scale such that Aj is vj/vk times preferred to Ak, 1j,kn. The overall preference vector v=[v1,v2,…,vn] is unique up to a positive scaling constant and we write v=[u1,u2,…,un] where can be any positive number. The relative ratios of v1,v2,…,vn constitute the overall results sought in the model without any specific unit of measure. For convenience, we use the normalization constraint v1+v2+…+vn=1 to get rid of and specify the overall preference values. For each criterion Cp (p=1,2,…,m), we assume that the local preferences x1p,x2p,…,xnp exist in ratio scale such that Aj is xjp/xkp times preferred to Ak under Cp, 1j,kn. Since the relative ratios xjp/xkp (1j,kn) are uniquely determined by the values in the local preference vector xp=[x1p,x2p,…,xnp] which can be arbitrarily scaled by any positive constant, we write xp=p[y1p,y2p,…,ynp] where p is a positive number. We note that, even when there exists original natural measurement for the performance of A1,A2,…,An under Cp, the local preference values x1p,x2p,…,xnp need not have linear functional relationship with these natural measurements. However, if the local preference values are expressed in well established unit of measure u (e.g. u=$1,000 or u=kg), then we write xp=pu[y1p,y2p,…,ynp]. As 1,2,…,m are arbitrary positive numbers, the values of pyjp and qyjq are arbitrary for criteria Cp and Cq. In particular, comparing the magnitudes of pyjp and qyjq under different criteria Cp and Cq is not sufficient because they do not have a common unit of
Ratio Scale Multi-Criteria Decision Making
measurement. Also, for each j (j=1,2,…,n), the alternative Aj is not fully described by [1yj1,2yj2,…,myjm] without some knowledge of the units of measure. We assume that criteria weights w1,w2,…,wm exist in ratio scale such that the weight of Cp is wp/wq times the weight of Cq, 1p,qm. The criteria weight vector w=[w1,w2,…,wm] is unique up to a positive scaling constant and we write w=[w1 ',w2',…,wm'] where can be any positive number. For convenience, we assume that has been chosen such that w1+w2+…+wm=1. The scaling factors 1,2,…,m and their relationship with the criteria weights have caused much ambiguity and confusion in Analytic Hierarchical Process (AHP) type models [4, 5, 11, 12]. The local preference values in xp are represented as pu[y1p,y2p,…,ynp] but the role of unit is obscure because the unit of measure u used in measuring the local preference values is usually not explicitly specified and p is an arbitrary positive number. In most AHP type models, 1,2,…,m are not explicitly shown as arbitrary positive numbers. This obscurity has led to some misunderstanding [12]. A common technique used to determine 1,2,…,m and hence fixing the numerical values of xp is to impose some artificial normalization constraint such as x1p+x2p+…+xnp=1. Even though x1p+x2p+…+xnp=1 is most frequently used, it is not the only one that can be used to specify the scaling constant p. Other normalization constraints such as xjp=1 for some j may be used as in the linking Pin AHP [9]. However, the DM must be aware of the role of the normalization constraint in the questioning procedure of the model. In this paper, we shall show the scaling factors 1,2,…,m explicitly in the model development until the units for measuring the local preferences are adequately specified and used in model calculations. We now give closer scrutiny to the role of scaling constant p by considering the prices in Canadian dollars C$200, C$300 and C$400 of three objects. The prices are represented by pu[2,3,4] where u="C$1" and p=100. Prices in different currencies are represented by different p and u which depend on the exchange rate that is applied. For pairwise evaluations of the relative ratios of the prices of these three objects, no particular currency u has to be specified and it is sufficient to use p[2,3,4] without specifying p and u is ignored. However, it is impossible to evaluate the impact of the prices of these objects solely from the knowledge of p[2,3,4] which captures only the relative ratios of the prices. The knowledge of one of the prices in a known currency is absolutely necessary to recover all the actual prices. It is important to realize the obvious fact that "the relative prices in p[2,3,4]" does not contain the crucial information pu=C$100 or "the price of the first object is C$200". p[2,3,4] can also represents the relative prices of [C$2,C$3,C$4] and thus it can not be used as a complete representation of [C$200,C$300, C$400]. Put it in another way, p[2,3,4] are pure numbers – they do not have complete information in terms of some well established unit of measure. For the derivation of criteria weights, the actual prices are needed to determine the impact of the price attributes on the overall preferences. When the actual prices are unknown, the importance of the actual prices in deriving the criteria weights can be emphasized by pu[2,3,4] or u[1,1.5,2] where p=1/2 and u="price of object 1".
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Unlike the currencies that are tangible and transparent to the DM, it is more challenging when dealing with local preferences which are intangible measures with no simple choice of p. It is vital to note that p[y1p,y2p,…,ynp] contain the relative ratios only and do not contain the original performances of A1,A2,…,An under Cp which are necessary for deriving the overall preferences of A1,A2,…,An. This implies that the pairwise matrices in AHP type models do not contain the crucial information of the original performances necessary for evaluating the alternatives with respect to all criteria. To rectify this, the criteria weights w1,w2,…,wm must capture and transform the original performances of A1,A2,…,An under Cp, p=1,2,…,m, into overall preferences. This requirement must be clearly stated in the questioning procedures for eliciting the criteria weights from the DM. In order to focus only on the more fundamental properties of the aggregation rules, we have assumed that all the relevant criteria are already identified and properly formulated. It follows from the autonomy of the criteria that Aj is always xjp/xkp time preferred to Ak (1j,kn) with respect to Cp regardless of whether any other alternative is deleted or added to A. In this paper, we compare the advantages and limitations of additive and multiplicative aggregation of x1,x2,…,xm with w based only on their fundamental mathematical properties. We do not consider the different impacts of numerous distinct methods proposed for the auxiliary problems of deriving the local priorities x1,x2,…,xm and the criteria weights in w. 4. LITERATURE REVIEW Local preferences under individual criteria are the basic elements used in most MCDM models. The criteria weights are usually modeled in ratio scale which allows an arbitrary positive scaling constant [11]. Local preferences are assumed to be measured in ordinal scale in ranking models [6], in interval scale in multiple attribute utility models [7] and in ratio scale in AHP type models [8, 9]. There are many methods proposed for deriving local preferences from pairwise comparison matrices [4, 5, 13-16]. The local preference vectors x1,x2,…,xm and the criteria weight vector w are then aggregated into overall preference vector v=[v1,v2,…,vn] by the additive aggregation [9, 17] or the multiplicative aggregation [2]. Goal programming and data development analysis are also used in some MCDM models [14, 16, 18, 19]. Most of these methods have been extensively analyzed and compared [1, 4, 5, 15, 20]. There are conflicting viewpoints and disagreements in many aspects of these MCDM models. The criteria weight wp can have many different interpretations. It is problematic when the generic term "criteria weight of importance" is used in questioning the DM without explaining its specific intended meaning. The criteria weights in w elicited from the DM may not be consistent with the associated mathematical operations used in the model [11, 12]. The local preferences x1,x2,…,xm are subjective evaluations from the DM with reference to the performance of the alternatives under individual criteria. Even when there are convenient original performance measures of the alternatives such as the prices in dollars, the subjective local preferences need not be linearly related. To simplify these subjective evaluations, different discrete intensity scales of 1-9 with verbal judgment are
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framed in different models for the DM to select one of the preference intensities [2, 17]. There is also disagreement on the validity of rank preservation of alternatives after some alternative is added or deleted [3, 21]. Another disagreement is whether additive or multiplicative aggregation is the better way to aggregate [22]. In the remainder of this paper, we compare the fundamental properties of the additive and multiplicative aggregation rules for ratio scales MCDM. 5. ADDITIVE AND MULTIPLICATIVE AGGREGATION RULES In Additive aggregation model, the overall preferences v1,v2,…,vn of A1,A2,…,An are estimated by the weighted Arithmetic means f1,f2,…,fn where
fj = w11y j1 + w 2 2 y j2 + ..... + w m m y jm , j = 1,2,...,n
(1)
In multiplicative aggregation model, the overall preferences v1,v2,…,vn of A1,A2,…,An are estimated by the weighted Geometric mean g1,g2,…,gn where
g j = (1y j1 )w1 ( 2 y j2 )w 2 .....( m y jm )w m , j = 1,2,...,n
(2)
The scaling constants 1,2,…,m are included in (1) and (2) to show their impact, or lack thereof, on the overall preferences explicitly. While 1,2,…,m must be explicitly specified in (1), it will be shown later that 1,2,…,m can be deleted in (2) with no change in the relative ratios of g1,g2,…,gn. However, since 1,2,…,m are associated to units of measure, they must be known and specified before determining the criteria weights w1,w2,…,wm. Mathematically, there is a one-one correspondence between the result f=[f1,f2,…,fn] from the additive model and the result of the multiplicative model applied to the exponentially transformed local preferences. Specifically, by taking exponential of (1), we have
e j = (e f
1y j1 w1
) (e
2 y j2 w 2
) .....(e
m y jm w m
) , j = 1,2,...,n
(3)
Thus the additive overall preference vector f=[f1,f2,…,fn] can be derived from the equations in (3) which is the multiplicative aggregation of the exponentially transformed local preferences. Thus, the same answer can be derived from (1) and (3). In similar fashion, the multiplicative overall preference vector g=[g1,g2,…,gn] in (2) can be derived from (4) ln (g j ) = w1ln (1y j1 ) + w 2ln ( 2 y j2 ) + ..... + w mln ( m y jm ), j = 1,2,...,n
(4)
Thus the result of the additive model can be obtained from the multiplicative model and vice versa. However, the additive and multiplicative aggregation models in equations (1) and (2) yield different results for the same data. Accordingly, they are regarded as different MCDM models, particularly due to the many very different interpretations of the intermediate terms wpxjp, xjp^(wp) and the criteria weights w1,w2,…,wm. This leads to the need for different questioning procedures to match the associated mathematical operations used in each model and different instructional protocols to ensure that the questioning procedures are well understood by the DM [1, 9].
6. FUNDAMENTAL PROPERTIES OF ADDITIVE AND MULTIPLICATIVE AGGREGATION RULES We now consider the fundamental properties (FP) of the additive model (FPA) and the multiplicative model (FPM). These properties are not affected by ancillary peripheral procedures such as interactive elicitations from decision makers, pairwise comparisons, and calculations of local priorities and criteria weights. (1) Partial Values (FPA1) In the additive model, wppyjp is a partial value of Aj under Cp. The right hand side of (1) is the sum of w11yj1, w22yj2,…,wmmyjm which are ratio scale measurements in commensurate units of overall preferences. Thus the w11yj1, w22yj2,…,wmmyjm components can be regarded as partial values in overall preference units. The partial value wppyjp represents the portion of the overall preference of Aj contributed from the criterion Cp. The estimated overall preference fj of Aj is the sum of all these partial values. (FPM1) In the multiplicative model, it is difficult to interpret (pyjp)^wp, p=1,2,…,m. The right hand side of (2) is a product of complicated terms which have no tractable units of measure. The contribution of Aj under Cp is not explicitly identified in (2). As factors of a product, the (1yj1)^w1, (2yj2)^w2,…, (myjm)^wm components are not portions that come together as gj. It is a mistake to treat (1yj1)^w1, (2yj2)^w2,…, (myjm)^wm as partial values measured in commensurate units and it is meaningless to compare their magnitudes. (2) Criteria Weights (FPA2) In the additive model, the criterion weights w1,w2,…,wm are conversion factors. The local preferences 1yj1,2yj2,…,myjm are converted to partial values w11yj1,w22yj2,…,wmmyjm. These partial values are in commensurate units and sum to fj which is an estimation of the overall preference vj of Aj. The criteria weights depend on the performances of the n alternatives with respect to each of the m criteria. When p is specified by imposing the normalization constraint py1p+py2p+…+pynp=1 as in the conventional AHP [11, 17], then wp=wppy1p+wppy2p+…+wppynp represents one unit equivalence of local preference in Cp measured in units of overall preference. This is a difficult concept because it may not be easy to conceive an alternative (real or imaginary) that achieves a local preference of py1p+py2p+…+pynp=1 under Cp. When pyjp=1 is imposed to specify p as in Linking Pin AHP [9], then wp=wppyjp represents the equivalence of the performance of Aj in Cp measured in units of overall preference. (FPM2) In the multiplicative model, there is no simple interpretation for criteria weights. It is difficult to interpret (pyjp)^wp in (2). The criteria weight wp, used as an exponent, does not have any direct apparent practical interpretation.
Ratio Scale Multi-Criteria Decision Making
The Open Operational Research Journal, 2008, Volume 2
Looking at the logarithmic transformed values of (2) as given in (4), it is clear that the criterion weights w1,w2,…,w m are the conversion factors that convert the logarithmic transformed values of local preferences ln(1yj1),ln(2yj2),…,ln (myjm) into partial values w1ln(1yj1),w2ln(2yj2),…,wmln (myjm). These partial values are in commensurate units and sum to ln(gj). However, it is much more complicated than the conversion factors in the additive model due to the logarithmic transformation. (3) Scaling Factors and Normalization Constraint (FPA3) In the additive model, the criteria weights w1,w2,…,wm are dependent on the normalization constraints and the associated scaling factors of 1,2,…,m. The units of measure for local preferences must be specified before they can be meaningfully converted to units of overall preferences. The local preferences 1yj1,2yj2,…, myjm are specified by fixing the unit of measure via 1,2,…,m with some normalization constraints. Then the criteria weights w1,w2,…,wm, as conversion factors, can be meaningfully elicited from the DM. We note that after w1,w2,…,wm are determined, the normalization constraint and the scaling factors 1,2,…,m can not be arbitrarily changed without altering the unit of measure of the local priorities. If a different normalization constraint is used and p is changed to p', then the criteria weight wp must be adjusted to wp'=wpp/p' to maintain the same results generated from the original units. Otherwise, unwarranted rank reversals may occur [12]. (FPM3) In the multiplicative model, the scaling factors 1,2,…,m do not appear explicitly in the relative ratios gj/gk. Thus gj/gk is independent of 1,2,…,m. as
The multiplicative model (2) can be written equivalently
g j = (1w1 2 w2 ..... m wm )(y j1w1 y j2 w2 .....y jm wm ), j = 1,2,...,n
(2a)
The term
1w1 2 w2 ..... m wm on the right hand side of (2a) is a common factor which can be deleted in the ratios gj/gk, 1j,kn. The relative ratios of g1,g2,…,gn are unchanged when the scaling factors 1,2,…,m and the normalization constraints are changed. The criteria weights w1,w2,…,wm are dependent on the actual performances underlying the local preference values. This is easily neglected since 1,2,…,m do not appear explicitly in the expressions for the relative ratios gj/gk (j,k=1,2,…,n). The DM is usually asked to provide w1,w2,…,wm as "relative criteria importance" without additional guidance. Even though it is complicated, correct guidance can be discerned from equation (4) and the criteria weights are converting factors that convert the logarithm of local priorities into commensurate units. (4) Criteria Importance (FPA4) In the additive model, there exists special form of normalization constraint for the criteria weights w1,w2,…,wm to be interpreted as weights of importance.
5
For each criterion Cp (p=1,2,…,m), xp=pu[y1p,y2p,…,ynp] where y1p,y2p,…,ynp are some specified local preferences under Cp, p is a positive number and u specifies an unit of measure if available. Let tp be a typical value of all local preferences y1p,y2p,…,ynp under Cp. In practice, tp is some standard value or aspiration value elicited from the DM or, it may simply be the median or average of y1p,y2p,…,ynp. By imposing the normalization constraint ptp=1, we can say that wp =wpptp is the amount of overall preference equivalent to a typical local preference under Cp. Interpreted as the contribution of a typical local preference, wp can be regarded as a measure of criteria importance of Cp [11]. (FPM4) In the multiplicative model, suppose Aj and Ak differ only in Cp and gj/gk jk, then the criteria weight wp is approximately ln(jk)/ln(yjp/ykp). It follows from (2a) that
gj gk
=
y jp
wp
y kp
wp
and thus
y jp
wp
y kp
wp
jk .
Taking logarithm, we get
wp
( jk ) y jp ( ) y kp
It is conceivable that the DM can compare two (real or imaginary) alternatives which differ only in one criterion Cp and provide jk as an estimated value of gj/gk. Then wp is approximately ln(jk)/ln(yjp/ykp). This provides a method for deriving criteria weights from the original performance of alternatives. It can also be used to check the validity of the model results. Unfortunately, there is no apparent insightful interpretation for ln(jk)/ln(yjp/ykp). (5) Local Level Dependency (FPA5) In the additive model, the ratio fj/fk depends on the magnitudes of yjp and ykp, even when yjp=ykp. Example 1. C={C1,C2}, A={A1,A2,A3,A4}, 1=2=1, w=[0.25,0.75], x1=[8,8,1,1] and x2=[6,1,6,1]. Table 2.
fj/fk Depends on the Magnitudes of yj1 and yk1 with yj1=yk1 C1
C2
Overall fj
alternative A1
8
6
6.5
alternative A2
8
1
2.75
alternative A3
1
6
4.75
alternative A4
1
1
1
Criteria Weight
0.25
0.75
f1/f2 = 6.5/2.75 = 2.3636 and f3/f4=4.75. As can be seen, the same local preferences in C1 do not cancel with each other. The difference in the ratios f1/f2 and f3/f4 is 2.3864, caused by the different magnitudes of local preferences 8 and 1 under C1.
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(FPM5) In the multiplicative model, the ratio gj/gk does not depend on the magnitudes of yjp and ykp, when yjp=ykp. m
m
g j = (y j1w1 )( y jp p ), g k = (y k1w1 )( y kp p ) w
p=2
m
gj gk
=
y
jp
y
kp
w
7. SUMMARY AND CONCLUSION and
p=2
wp
p=2 m
wp
.
p=2
Suppose yj1=yk1. Then we can see that the ratio gj/gk is independent of the magnitude yj1. Example 2. C={C1,C2}, A={A1,A2,A3,A4}, 1=2=1, w=[0.25,0.75], x1=[8,8,1,1] and x2=[6,1,6,1]. Table 3. g j/gk Independent of the Magnitudes of yj1 and yk1 with yj1=yk1 C1
C2
Overall gj
alternative A1
8
6
6.4474
alternative A2
8
1
1.6818
alternative A3
1
6
3.8337 1
alternative A4
1
1
Criteria Weight
0.25
0.75
preference values. Adding or deleting an alternative does not alter the perception of the criteria.
g1/g2 = g3/g4 = 3.8337. Same local preferences in C1 are canceled with each other. Thus the different magnitudes of the local preferences 8 and 1 under C1 do not lead to difference results for the ratios g1/g2 and g3/g4. However, the huge difference between the magnitudes 8 and 1 under C1 is expected to have significant impact as in (FPA5) rather than absolutely no impact in (FPM5). (6) Addition or Deletion of Alternative (FPA6) In the additive model, if the values of 1,2,…,m are specified by the normalization x1p+x2p+…+xnp=1 (p=1,2,…,m), then the criteria weights w1,w2,…,wm need to be adjusted when some alternative is added or deleted in order to maintain commensurability. If linking pin normalization pyjp=1 is used, the criteria weights w1,w2,…,wm remain unchanged when some alternative is added or some alternative other than the j-th one is deleted. When an alternative is added or deleted, the terms in the left hand side of the normalization constraint py1p+py2p+…+pynp=1 is changed and p attains a new value p'. The partial value of Aj under Cp is given by wppyjp and also by wp'p'yjp. Thus wp'=wpp/p' and the criteria weight wp needs to be adjusted to wp'. (FPM6) In the multiplicative model, the scaling factors 1,2,…,m do not appear explicitly in the relative ratio gj/gk. The criteria weights w1,w2,…,wm may remain unchanged when some alternative is added or deleted The criteria weights w1,w2,…,wm are dependent on the actual performances of the alternatives underlying the local
The advantages and limitations of the additive and multiplicative aggregation rules are explored and summarized in Table 4. The additive model has the useful interpretation of wppyjp as partial value of the performance of Aj under Cp and the simple interpretation of the criteria weights as conversion factors. Incorporating these interpretations would increase the likelihood that the questioning procedures are better understood by the DM in practice. The partial values in the additive model are converted by the scaling factors (criteria weights) to commensurate values which are summed to the overall preference values f1,f2,…,fn. For the additive version of the multiplicative model (equation in (4)) the partial values must undergo the logarithmic transformations before conversion by the scaling factors (criteria weights). The implication is that the weights have to be derived differently. In practice, this is not done. Instead, weights are generated in a similar manner for both methods. The multiplicative adherents [2, 22] criticize the ambiguous criteria weights used in AHP. Yet, they themselves do not pay attention to the wording necessary for the criteria weights in the multiplicative model. In the additive versions of both models, commensurate values must be generated before any addition takes place. The necessity of commensurate units defines the role of criteria weights as conversion factors. 1,2,…,m do not appear explicitly in the expressions for the relative ratios gj/gk (j,k=1,2,…,n). Thus the relative ratios are unchanged when the scaling factors 1,2,…,m and the normalization constraints are changed or some alternative is added or deleted. However, this leads to the common error of neglecting the actual performances of the alternatives underlying the local preference values in the derivation of criteria weights in the multiplicative model which is misconstrued as an advantage of having fewer elements to manipulate. In actual fact, it is a weakness of the multiplicative model. This is similar to deriving criteria weights w1,w2,…,wm from the prices of three objects given only as [2,3,4] without the crucial knowledge of the actual prices [C$200,C$300, C$400]. This means the performances of the alternatives are not fully used in the multiplicative model. Furthermore, it is not logical that the ratio gj/gk is independent of the magnitudes of yjp and ykp when yjp and ykp. are equal or have similar magnitudes. Thus the results of the overall preference values from the multiplicative model are not as credible as the results from the additive model. Also it is more complicated to estimate the criteria weight wp from ln(jk)/ln(yjp/ykp) (p=1,2,…,m) where jk is elicited from the DM's comparison of two imaginary alternatives which differ only in Cp with local preferences pyjp and pykp. It is evident that the additive aggregation model, based on the six fundamental properties (FPA1)-(FPA6), is superior insofar as it is effective and easier for the DM to understand and use. Proper interpretations and careful handling of the normalization constraints used in the model are essential elements to be incorporated properly into the questioning procedures. The criteria weights used in the multiplicative
Ratio Scale Multi-Criteria Decision Making
Table 4.
The Open Operational Research Journal, 2008, Volume 2
Fundamental Properties of Additive and Multiplicative Rules Additive Model
Multiplicative Model
Partial Values
wppyjp is part of overall value of Aj from Cp in units of overall preferences
(pyjp)^w p is not partial value of Aj No tractable units of measurement
Criteria Weights
Criteria weights are conversion factors, depend on alternatives' performances
Criteria weights have no practical interpretation, no explicit relation with alternatives' performances
Scaling Factors
1,2,...,m fixed before deriving w. Criteria weights need to be adjusted for changes in 1,2,...,m
1,2,...,m are arbitrary after deriving w. Relative ratios of g1,g2 ,...,gn are independent of 1,2,...,m
Normalization Constraint
Criteria weights depend on the normalization constraints
1,2,...,m do not appear explicitly in the expression for gj/gk. The relative ratios gj/gk are independent of 1,2,...,m
Criteria Importance
Criteria weights may be interpreted as weights of importance
Criteria weight w p are possibly an approximation of ln()/ln(yjp/ykp)
Local Level Dependency
fj/fk depends on the magnitudes of yjp and ykp when yjp=ykp.
gj/gk independent of the magnitudes of yjp and ykp when yjp=ykp.
Add/Delete Alternative
For sum to one normalization, criteria weights adjusted when alternative is added or deleted For linking pin normalization, criteria weights unchanged when alternative is added or deleted
1,2,…,m do not appear explicitly in the relative ratio gj/gk . Criteria weights unchanged when alternative is added or deleted
aggregation have complicated meanings which are not well understood and often mixed up in the ambiguous notion of "criteria importance". Furthermore, the relative ratios in the multiplicative aggregation do not depend on similar local preference values which cancel each other out mathematically. Thus we recommend the additive aggregation rule over the multiplicative aggregation rule.
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Received: October 19, 2007
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Revised: January 2, 2008
Accepted: January 30, 2008
